Download periodic sequences of numbers in generalized arithmetic and

Palestine Journal of Mathematics
Vol. 4(1) (2015) , 164–169
© Palestine Polytechnic University-PPU 2015
PERIODIC SEQUENCES OF NUMBERS IN GENERALIZED
ARITHMETIC AND GEOMETRIC ALTERNATE
PROGRESSIONS
Julius Fergy T. Rabago
Communicated by Ayman Badawi
MSC 2010 Classifications: Primary 11B25; Secondary 11Y55.
Sequence of numbers with alternate common difference and ratio, general term an , periodic sequence, sum.
Abstract The paper provides a generalization of the arithmetic-geometric alternate sequence
introduced recently by Rabago [2].
1 Introduction
The natural numbers, usually denoted by N, is given by the sequence 1, 2, 3, 4, 5, 6, 7, . . .. This
type of number sequence is an example of what we call arithmetic sequence. An arithmetic sequence is a number sequence in which every term except the first is obtained by adding a fixed
number, called the common difference, to the preceeding term. Another example is the sequence
1, 3, 5, 7, 9, 11, . . . whose common difference is 2. Denote the nth term of the arithmetic sequence with first term a and common difference d as an and the sum of the first n terms of the
sequence as Sn . Then, an is define recursively as
a1 = a,
an = an−1 + d, (n ≥ 2).
An explicit formula for an is given by
an = a + (n − 1)d, (n ≥ 2).
The sum Sn is given by
n
[2a + (n − 1)d], (n ≥ 1).
2
Another type of sequence of numbers is the so-called geometric sequence. A geometric
sequence is a number sequence in which every term except the first is obtained by multiplying the
previous term by a constant, called the common ratio. For example, 2, 4, 8, 16, . . . is a geometric
sequence with common ratio 2. Let an denote the nth term of the geometric sequence with first
term a and common ratio r. Then, an is define recursively as
Sn =
a1 = a,
an = an−1 · r,
(n ≥ 2).
An explicit formula for an is given by
an = a · rn−1 , (n ≥ 2).
The sum Sn is given by
rn − 1
, r 6= 1 (n ≥ 1).
r−1
In a recent paper, Rabago [2] introduced the concept of arithmetic-geometric alternate sequence of numbers as follows:
Sn = a
Definition 1.1. A sequence of numbers {an } is called an arithmetic-geometric alternate sequence
of numbers if the following conditions are satisfied:
a2k
= r,
(i) for any k ∈ N,
a2k−1
(ii) for any k ∈ N, a2k+1 − a2k = d,
where r and d are called the common ratio and common difference of the sequence {an }, respectively.
In this study, we present two types of generalization of the arithmetic-geometric alternate
sequence [2]. We also present in this work an explicit formula for the nth term of the sequence
as well as the sum for the first n terms.
PERIODIC SEQUENCES OF NUMBERS
165
2 Periodic Arithmetic-Geometric Alternate Sequence
We start off with the definition of what we call periodic sequence of numbers with alternate
common difference and ratio.
Definition 2.1. A sequence of numbers {an } is called a periodic sequence of numbers with
alternate common difference and ratio if for a fixed natural number m the following conditions
are satisfied:
(i) for any k = 1, 2, . . . and for all natural number j ≤ m − 1,
am(k−1)+j +1 − am(k−1)+j = d,
(ii) for any k = 1, 2, . . .,
amk+1
= r.
amk
Clearly, the above definition takes the following form:
a1 , a1 + d, a1 + 2d, . . . , a1 + (m − 1)d, (a1 + (m − 1)d)r, (a1 + (m − 1)d)r + d, . . . ,
(a1 + (m − 1)d)r + (m − 1)d, ((a1 + (m − 1)d)r + (m − 1)d)r, . . .
(2.1)
From the previous definition we may define m as the period of the sequence and the terms
{a1 , a2 , . . . , am } can be defined as the elements of the 1st interval (or period) of length m,
{am+1 , am+2 , . . . , a2m } as the elements of the 2nd interval of length m, and so on, and in general, the terms {a(k−1)m+1 , a(k−1)m+2 , . . . , akm } can be considered as the elements of the k -th
interval of length m. It can be observed easily that for each interval, the terms are in arithmetic
progression with d as the common difference.
Throughout in the paper we denote the greatest integer contained in x as bxc.
Theorem 2.2. Let d and r be any two real numbers such that r 6= 1 and {an } be a periodic
sequence of numbers with alternate common difference d and ratio r. Then, the formula for the
nth term of {an } is given by,
1 − r e1
e1
an = a1 r + (m − 1)
dr + (n − 1 − me1 )d,
(2.2)
1−r
1
where e1 = n−
m .
Proof. The formula is clearly true for n ≤ m. We only have to show that the formula is valid
for n > m. To do this, first, we will show that formula (2.2) holds for any fixed natural number
k > 1. We let k be a fixed natural number and p = m(k − 1) + j , where j is a natural number
less than m. Note that ap+1 = ap + d for all j ≤ m − 1. This implies that,
1 − r e1
dr + (p − 1 − me1 )d + d
ap = a1 re1 + (m − 1)
1−r
j
k
1
Here, e1 = p−
m . Replacing p by m(k − 1) + j , we’ll obtain,
ap = a1 r
Because
k−1
1 − rk−1
+ (m − 1)
1−r
dr + jd.
m(k − 1) + j − 1
m(k − 1) + j
=
,
m
m
for all natural number j ≤ m − 1, then
1 − r e0
ap+1 = a1 r + (m − 1)
1−r
j
k
−1
where e0 = (p+1)
.
m
e0
dr + ((p + 1) − 1 − me0 )d,
166
Julius Fergy T. Rabago
Now we need to show that amk+1 = am k · r for each interval k . Clearly, amk+1 = amk · r is
true for k = 1. So, we assume that amp+1 = amp · r for some natural number p > 1. Hence,
1 − rp
am(p+1) · r =
a1 rp + (m − 1)
dr + (m(p + 1) − 1 − mp)d · r
1−r
1 − rp
p+1
= a1 r
+ (m − 1)
dr2 + (m − 1)dr
1−r
1 − rp+1
dr
= a1 rp+1 + (m − 1)
1−r
= am(p+1)+1 ,
proving the theorem.
Lemma 2.3. For any integer m > 0 and natural number n,
j k
n X
m n+m
n
i
n+1−
=
.
m
m
2
m
i=1
Lemma 2.4. For any integer m > 0 and natural number n,
n
X
rei = m − 1 + rm
i=1
where ei =
i
m
1 − r e n −1
1−r
+ (n + 1 − men ) ren , (r 6= 1),
.
For the proof of Lemma (2.3) and Lemma (2.4), see [2] and [3], respectively.
Theorem 2.5. The sum of the first n terms of (2.1) is given by
Sn = nM + (a1 − M )Rn +
n(n − 1)d
− mdEn ,
2
(2.3)
where
M
(m − 1)dr
1−r
=
1 − ren −1
= m − 1 + rm
+ (n − men ) ren ,
1−r
n−1
,
=
m
n−1
m n+m−1
=
n−
.
m
2
m
Rn
en
En
Proof. Let
m1 > 0 be an integer, r be a real number different from 0 and 1, n a natural number,
and ei = i−
m . Let {an } be a sequence of the form as in (2.1). Then,
n
X
ai
=
i=1
=
n X
1 − rei
dr + (i − 1 − mei )d
a1 r + (m − 1)
1−r
i=1
n
n
X
n(m − 1)dr
(m − 1)dr X ei n(n − 1)d
+ a1 −
r +
− md
ei
1−r
1−r
2
ei
i=1
i=1
and by Lemma (2.3) and Lemma (2.4), conclusion follows.
We end this section with the following remark.
Remark 2.6. We note that by letting m → ∞ in (2.2), we’ll obtain the explicit formula for
the usual arithmetic sequence of numbers with common difference d. Also, one may verify that
Rn → n as m → ∞ and that the formula for the sum of n terms Sn given by (2.3) in Theorem
1)
(2.5) will approach a1 n + n(n−
d as m → ∞.
2
PERIODIC SEQUENCES OF NUMBERS
167
3 Periodic Geometric-Arithmetic Alternate Sequence
In this section, we present another generalization of arithmetic-geometric sequence with the following definition of a periodic sequence of numbers with alternate common ratio and difference.
Definition 3.1. A sequence of numbers {an } is called a periodic sequence of numbers with alternate common ratio r and difference d if for a fixed natural number m the following conditions
are satisfied:
(i) for any k = 1, 2, . . . and for all natural number j ≤ m − 1,
am(k−1)+j +1
= r,
am(k−1)+j
(ii) for any k = 1, 2, . . ., amk+1 − amk = d.
It can be seen easily that the number sequence {an } has the following form:
a1 , a1 r, a1 r2 , . . . , a1 rm−1 , a1 rm−1 + d, (a1 rm−1 + d)r, (a1 rm−1 + d)r2 , . . . ,
(a1 rm−1 + d)rm−1 + d, ((a1 rm−1 + d)rm−1 + d)r, . . .
(3.1)
st
Here we say that the terms {a1 , a2 , . . . , am } belong to the 1 interval of length m, {am+1 , am+2 , . . . , a2m }
belong to the 2nd interval of length m, and so on, and in general, the terms {a(k−1)m+1 , a(k−1)m+2 , . . . , akm }
belong to the k -th interval of length m. Note that for each interval, the terms are in geometric
progression with r as the common ratio.
Theorem 3.2. Let d and r be any two real numbers such that r 6= 1 and {an } be a periodic
sequence of numbers with alternate common ratio r and difference d. Then, the formula for the
nth term of {an } is given by,
1 − (rm−1 )e1
an = a1 rn−1−e1 + d
rn−1−me1 ,
(3.2)
1 − rm−1
1
where e1 = n−
m .
Proof. Obviously formula (3.2) is valid for every natural number n ≤ m. We only need to
verify the validity of the formula for n > m. To do this, we first show that for every interval
k = 1, 2, . . ., the formula is true and then, we show that for every k , amk+1 = amk + d.
Now, let p = m(k − 1) + j with k fixed then, ap+1 = ap · r for all natural number j ≤ m − 1.
Hence,
1 − (rm−1 )e1
p−1−me1
p−1−e1
r
· r,
ap+1 = a1 r
+d
1 − rm−1
j
k
1
where e1 = p−
. Simplifying and noting that
m
m(k − 1) + j
m(k − 1) + j − 1
=
,
m
m
for all natural number j ≤ m − 1, we obtain
ap+1 = a1 r(p+1)−1−e0 + d
1 − (rm−1 )e0
1 − rm−1
r(p+1)−1−me0 ,
j
k
−1
where e0 = (p+1)
. On the other hand, it can be shown easily that amk+1 = amk + d is true
m
for k = 1. So, we assume that amp+1 = amp + d for some natural number p > 1. This implies
that,
1 − (rm−1 )e1
am(p+1) + d = a1 rm(p+1)−1−e1 + d
rm(p+1)−1−me1 + d
1 − rm−1
j
k
j
k
−1
−1
where e1 = m(p+1)
. But, m(p+1)
= p, then
m
m
1 − (rm−1 )p m(p+1)−1−mp
r
+d
am(p+1) + d = a1 rm(p+1)−1−p + d
1 − rm−1
1 − (rm−1 )p m−1
= a1 r(m−1)(p+1) + d
r
+
1
1 − rm−1
1 − (rm−1 )p+1
(m−1)(p+1)
= a1 r
+d
1 − rm−1
= am(p+1)+1 .
168
Julius Fergy T. Rabago
This proves the theorem.
Similar to what we remarked in the previous section, we can notice easily that formula (3.2)
will approach the form a1 rn−1 as m → ∞. That is, we’ll obtain the explicit formula for the
usual geometric sequence of numbers with common ratio r.
Lemma 3.3. Let R be the sum
n
X
ri−1−ei ,
r 6= 0, 1,
i=1
where ei =
i−1 m
. Then, for any natural numbers m and n,
R=
where p =
n−1 m
1 − rm
1−r
1 − (rm−1 )p
1 − rm−1
+
1
rp
1 − rn−mp
1−r
,
(3.3)
.
Proof. Let
m1 > 0 be an integer, r be a real number different from 0 and 1, n a natural number,
. Then,
and p = n−
m
n
X
r
i−1−ei
=
i=1
n
X
r
i−1
i=1
=
(m
X
1
b i−
m c
1
r
ri−1 +
i=1
+
(
=
p−1
1
r
m
X
X
2 X
2m
3m
1
1
ri−1 +
ri−1 + . . .
r
r
i=m+1
i=2m+1

mp
n
 1 p X
X
ri−1
ri−1 +

r
i=mp+1
i=(p−1)m+1
ri−1 + rm−1
m
X
i=1
ri−1 + rm−1
m
2 X
i=1
+ rm−1
p−1
m
X
i=1
)
ri−1
ri−1 + . . .
+
i=1
1
rp
n−mp
X
ri−1
i=1
p
n−mp
1 − rm X m−1 j−1
1 X i−1
r
=
(r
)
+ p
1−r
r
i=1
j =1
1 − rm
1 − (rm−1 )p
1 1 − rn−mp
=
+ p
.
1−r
1 − rm−1
r
1−r
Lemma 3.4. Let R̄ be the sum
n
X
ri−1−mei ,
r 6= 0, 1,
i=1
where ei =
i−1 m
. Then, for any natural numbers m and n,
R̄ =
where p =
n−1 m
n−1
m
1 − rm
1−r
+
1 − rn−mp
1−r
,
(3.4)
.
We omit the proof since it similar on how we prove (3.3).
Theorem 3.5. The sum of the first n terms of (3.1) is given by
d
d
Sn = a1 −
R
+
R̄,
1 − rm−1
1 − rm−1
where R and R̄ are given by equations (3.3) and (3.4), respectively.
(3.5)
169
PERIODIC SEQUENCES OF NUMBERS
Proof. Let ei =
n
X
i=1
i−1 ai
m
n X
1 − (rm−1 )ei
i−1−mei
i−1−ei
r
=
a1 r
+d
1 − rm−1
i=1
X
X
n
n
d
d
i−1−ei
=
a1 −
+
r
ri−1−mei
1 − rm−1
1 − rm−1
i=1
i=1
, and by Lemma (3.3) and Lemma (3.4), conclusion follows.
Note that the formula given by (3.5) will approach the expression of the form a1
m → ∞ because R →
1−rn
1−r
1−rn
1−r
as
as m → ∞.
References
[1] T. Koshy, ‘‘Elementary number theory with applications’’, 2nd ed., Elsevier, USA, 2007.
[2] J.F.T. Rabago, ‘‘Arithmetic-geometric alternate sequence’’, Scientia Magna, 8 (2012), No. 2, 80-82.
[3] J.F.T. Rabago, ‘‘Sequence of numbers with three alternate common differences and common ratios’’, Int.
J. Appl. Math. Res., 1 (2012), No. 3, 259-267.
[4] K. H. Rosen, ‘‘Elementary number theory and its applications’’, Addison-Wesley Publishing Co., Massachusetts, 1986.
Author information
Julius Fergy T. Rabago, Department of Mathematics and Computer Science, College of Science, University of
the Philippines, Baguio Governor Pack Road, Baguio City 2600, PHILIPPINES.
E-mail: [email protected]
Received: July 22, 2013.
Accepted: December 22, 2013.